Let's say I can measure something with a know standard deviation of $\sigma$. Let's say I measure it $n$ times and take the mean of the $n$ measurements to determine a more accurate mean value. What is the new $\sigma$ for my mean measurement? I can intuitively know that it'll be smaller than the original $\sigma$ but still greater than $0$, but how does one calculate the new $\sigma$ value?
Bonus question (but much harder I suspect):
What if still have the $n$ measurements, but they all have slightly different $\sigma$. How do I now compute/derive the new $\sigma$ (given that the $n$ measurements all had slightly different $\sigma$). (In case it helps, $n$ will typically be ~$40$, but it can be as little as $10$ or as much as $500$; also, the different $\sigma$ do vary, but not by much: the smallest $\sigma$ and the biggest $\sigma$ will differ by at most ~$2\times$)